00:01
We are going to perform the indicated block multiplication of matrices, a and b.
00:07
Matrix a is 0100 to 1 -0 -0 -0 -0 -0 -0 -2, and matrix b is 1 -0 -0 -0 -0 -0 -3 and 0 -0.
00:20
And we are going to partition them like, in the case of matrix a like this in four sub -matrices, the left upper corner is a 2x2 matrix 01 to 1 the upper right is a null or zero matrix 0 0 0 order 2 the lower left is a vector of row vector of two elements 0 and the lower right corner is a vector 1 row 2 columns 1 2 so we have 4 some matrices and for v is similar but now the partition is this way we use these dot lines to indicate the partition so we have the upper left corner is a 2x2 -12 901 -1 the upper right is a column vector of two components 0 here the lower left is a 2x2 matrix 0 0 0 the 0 matrix and the zero matrix and the zero matrix and the 0 the lower right corner is column vector with two components three one.
01:45
So the idea here is if we have a partition of matrix a, let's say a block a11, another block a12 in such a way, these two blocks have the same number of rows, then we have two more blocks b21, it's a block with the same number of columns as a11, and then we have b22 with the same number of rows of b21 and the same number of columns as a12.
02:24
And we have another partition in b, let's say b11, b12, b21, b12, b2, for example, as we did up here.
02:40
The idea is that we can multiply a times b, in a blockwise way.
02:51
That is, if we imagine these two, these partitions of a and b were numbers, let's for the moment, let's suppose that.
03:01
That is, if this a were a 2x2 matrix and b were a 2 by 2 matrix, we know how to multiply them.
03:09
We know that matrix a, sorry, i made a mistake over here.
03:19
I used letter b is letter a.
03:25
I didn't notice that.
03:27
This is a here and b has b all the way.
03:33
Good.
03:33
So a to 1, a2, times the matrix b in the partition form, b11, b1, b12, b21 and b22.
03:53
If these were numbers, these were 2 by 2 matrices, both of them.
03:59
2 by 2a and 2 by 2b, and we know how to multiply that.
04:03
That will be a11 times b11 plus a12 times 3211.
04:18
That will be the lower, sorry, the upper left block or number in the resulting matrix.
04:27
Then we do the same for the second element in the first row will be a11, b12, b12, 112, plus a12 times b12 plus a1222 times b22222.
04:42
And now we do the other column, the other row, sorry, that will be a21 times b11 plus a22 times b21.
05:10
And that, the last element on the lower right corner will be a21, a2 times b12 plus a22 times b22.
05:32
Let me arrange this a moment.
05:40
Much here, okay, 2 -2.
05:45
That is, we have done here at 2x2 and 2 by 2 product of matrices.
05:53
But of course if a -1 -1, a -2, a -2, and the same 4b are blocks.
06:00
There is, all these are sub -matresses, not numbers.
06:05
Then we expect this work as well whenever we have the correct partition.
06:14
That is the products of matrices here the block products are well defined and of course the sum also are well defined that's the the only thing we require if all the multiplications here can be done without any problem and the sums then it's true that doing this multiplication block y this way is the same as multiplying the matrices without any partitions so let's see that.
06:49
You have this partition here, and this partition has the property that old products and sums here can be done.
06:58
Let's see that.
07:00
So a times b in our particular case will be matrix 01.
07:08
For the moment, i'm not going to use color.
07:11
0121.
07:13
It's faster 0, then 01212.
07:20
That's matrix a i put that points to that lines to see the partition let's verify that's correct times b which is 1 2 negative 1 1 1 0 0 0 and 0 0 and 3 1 and 3 1 and the partition is right here so let's do the block multiplication so we are going to indicate the block multiplication as we did here in general.
08:03
That is the block a11 is this one here times block b11.
08:13
That is 01 to 1 times 12 negative 1 plus 0000000 or 0 -0 -0 matrix times the 0 -0 matrix times the 0 -0 matrix.
08:33
I'm going to indicate it, even though we know it's going to be the zero matrix code, it's good to verify that we can do all multiplications and sums.
08:46
Okay, now we do the first two blocks in the upper, the first row, let's say, of the blocks times this second block here.
08:56
So we get 01 to 1 times this 1, 0 0, 0, plus here this 0 0 0 times this one here 3 1 now we do the other row let's say in the block multiplication so we have this times this metrics here so we get 0 in a row pattern times 1 2 negative 1 plus this one here 1 2 is a row vector times the 6 .2 is a row vector times the matrix 2 by 2 good and now we do the same with these two again but with the second row column of the block wise multiplication so we get 0 0 row vector times this block here 0 column vector plus 1 2 row vector times 1 3 here 3 1 column vector.
10:26
And now let's do the multiplications and sums that are indicated.
10:30
So here we have 2 by 2 times 2 by 2 matrix.
10:34
So the result is 2 by 2.
10:35
We get 0 times 0 is 0 plus 1 times 0 plus 1 is 901.
10:43
0 minus 1 is 901.
10:46
0 times 2 is 0 plus 1 is 1.
10:51
2 times 1 is 1.
10:52
2 times 1 is 2 minus 1 is 1.
10:55
2 times 1 is 1.
10:55
2 times 1 is 1.
10:55
2 minus 1 is 1.
10:56
And two times two plus one times one is five.
11:03
So we get this.
11:05
This is the two by two block resulting from this first product, and that can be added to this other product, which is a two by two times two by two again.
11:17
In this case, it's easy, it's zero, zero, zero...